Approximating subset kk-connectivity problems

A subset $T \subseteq V$ of terminals is $k$-connected to a root $s$ in a directed/undirected graph $J$ if $J$ has $k$ internally-disjoint $vs$-paths for every $v \in T$; $T$ is $k$-connected in $J$ if $T$ is $k$-connected to every $s \in T$. We consider the {\sf Subset $k$-Connectivity Augmentation} problem: given a graph $G=(V,E)$ with edge/node-costs, node subset $T \subseteq V$, and a subgraph $J=(V,E_J)$ of $G$ such that $T$ is $k$-connected in $J$, find a minimum-cost augmenting edge-set $F \subseteq E \setminus E_J$ such that $T$ is $(k+1)$-connected in $J \cup F$. The problem admits trivial ratio $O(|T|^2)$. We consider the case $|T|>k$ and prove that for directed/undirected graphs and edge/node-costs, a $\rho$-approximation for {\sf Rooted Subset $k$-Connectivity Augmentation} implies the following ratios for {\sf Subset $k$-Connectivity Augmentation}: (i) $b(\rho+k) + {(\frac{3|T|}{|T|-k})}^2 H(\frac{3|T|}{|T|-k})$; (ii) $\rho \cdot O(\frac{|T|}{|T|-k} \log k)$, where b=1 for undirected graphs and b=2 for directed graphs, and $H(k)$ is the $k$th harmonic number. The best known values of $\rho$ on undirected graphs are $\min\{|T|,O(k)\}$ for edge-costs and $\min\{|T|,O(k \log |T|)\}$ for node-costs; for directed graphs $\rho=|T|$ for both versions. Our results imply that unless $k=|T|-o(|T|)$, {\sf Subset $k$-Connectivity Augmentation} admits the same ratios as the best known ones for the rooted version. This improves the ratios in \cite{N-focs,L}

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios

Covering symmetric skew-supermodular functions with hyperedges

In this paper we give results related to a theorem of Szigeti that concerns the covering of symmetric skew-supermodular set functions with hyperedges of minimum total size. In particular, we show the following generalization using a variation of Schrijver’s supermodular colouring theorem: if p1 and p2 are skewsupermodular functions whose maximum value is the same, then it is possible to find in polynomial time a hypergraph of minimum total size that covers both of them. Note that without the assumption on the maximum values this problem is NP-hard. The result has applications concerning the local edge-connectivity augmentation problem of hypergraphs and the global edge-connectivity augmentation problem of mixed hypergraphs. We also present some results on the case when the hypergraph must be obtained by merging given hyperedges

Constrained Edge-Splitting Problems

Splitting off two edges su, sv in a graph G means deleting su, sv andadding a new edge uv. Let G = (V +s,E) be k-edge-connected in V(k >= 2) and let d(s) be even. Lov´asz proved that the edges incident to scan be split off in pairs in such a way that the resulting graph on vertexset V is k-edge-connected. In this paper we investigate the existence ofsuch complete splitting sequences when the set of split edges has to meetadditional requirements. We prove structural properties of the set of thosepairs u, v of neighbours of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type.By applying this method we obtain a short proof for a recent result ofNagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity (l-edge-connectivity, resp.) in V , provided d(s) >= 6. If k and l are both even then such a pair always exists. Using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size